I feel like if there is only one, like the question states, it has to be the first one, preimage resistance. However, given a hash $h = 000....0$
I can find a bunch of $m$'s that could give that particular hash. Maybe since there are lots of solutions you can't pin point the specific $m$ that gave that value?

I first thought it might've been supposed to be $h(m)=H(M_1)\oplus H(M_2)\oplus\dotsb\oplus H(M_\ell)$, where $H$ is a PRF, but that's not preimage resistant either. (You need to hash about as many random blocks as there are bits in the output and do some linear algebra to get first preimages, but that's a negligible amount of work.)
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Ilmari KaronenDec 16 '11 at 19:07